Rigorous justification of the short-pulse equation

We prove that the short-pulse equation, which is derived from Maxwell equations with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals.Comment: 15 pages, no figure

Single-Shot Visualization Of Evolving Laser- Or Beam-Driven Plasma Wakefield Accelerators

We introduce Frequency-Domain Tomography (FDT) for visualizing sub-ps evolution of light-speed refractive index structures in a single shot. As a prototype demonstration, we produce single-shot tomographic movies of self-focusing, filamenting laser pulses propagating in a transparent Kerr medium. We then discuss how to adapt FDT to visualize evolving laser-or beam-driven plasma wakefields of current interest to the advanced accelerator community. For short (L similar to 1 cm), dense (n(e) similar to 10(19) cm(-3)) plasmas, the key challenge is broadening probe bandwidth sufficiently to resolve plasma-wavelength-size structures. For long (L similar to 10 to 100 cm), tenuous (n(e) similar to 10(17) cm(-3)) plasmas, probe diffraction from the evolving wake becomes the key challenge. We propose and analyze solutions to these challenges.Physic

Nano-jet Related to Bessel Beams and to Super-Resolutions in Micro-sphere Optical Experiments

The appearance of a Nano-jet in the micro-sphere optical experiments is analyzed by relating this effect to non-diffracting Bessel beams. By inserting a circular aperture with a radius which is in the order of subwavelength in the EM waist, and sending the transmitted light into a confocal microscope, EM fluctuations by the different Bessel beams are avoided. On this constant EM field evanescent waves are superposed. While this effect improves the optical-depth of the imaging process, the object fine-structures are obtained, from the modulation of the EM fields by the evanescent waves. The use of a combination of the micro-sphere optical system with an interferometer for phase contrast measurements is described.Comment: 14 pages , 2 figure

Diffraction of Bloch Wave Packets for Maxwell's Equations

We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schr\"odinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations

Optical vortices in dispersive nonlinear Kerr type media

The applied method of slowly varying amplitudes of the electrical and magnet vector fields give us the possibility to reduce the nonlinear vector integro-differential wave equation to the amplitude vector nonlinear differential equations. It can be estimated different orders of dispersion of the linear and nonlinear susceptibility using this approximation. The critical values of parameters to observe different linear and nonlinear effects are determinate. The obtained amplitude equation is a vector version of 3D+1 Nonlinear Schredinger Equation (VNSE) describing the evolution of slowly varying amplitudes of electrical and magnet fields in dispersive nonlinear Kerr type media. We show that VNSE admit exact vortex solutions with classical orbital momentum $\ell=1$ and finite energy. Dispersion region and medium parameters necessary for experimental observation of these vortices, are determinate. PACS 42.81.Dp;05.45.Yv;42.65.TgComment: 24 page

Approximation of high-frequency wave propagation in dispersive media

We consider semilinear hyperbolic systems with a trilinear nonlinearity. Both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, and typical solutions oscillate with frequency proportional to $1/\varepsilon$ in time and space. Moreover, solutions have to be computed on time intervals of length $1/\varepsilon$ in order to study nonlinear and diffractive effects. As a consequence, direct numerical simulations are extremely costly or even impossible. We propose an analytical approximation and prove that it approximates the exact solution up to an error of $\mathcal{O}(\varepsilon^2)$ on time intervals of length $1/\varepsilon$. This is a significant improvement over the classical nonlinear Schrödinger approximation, which only yields an accuracy of $\mathcal{O}(\varepsilon)$

Improved error bounds for approximations of high-frequency wave propagation in nonlinear dispersive media

High-frequency wave propagation is often modelled by nonlinear Friedrichs systems where both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, which causes oscillations with wavelengths proportional to $\varepsilon$ in time and space. A prominent example is the Maxwell--Lorentz system, which is a well-established model for the propagation of light in nonlinear media. In diffractive optics, such problems have to be solved on long time intervals with length proportional to $1/\varepsilon$. Approximating the solution of such a problem numerically with a standard method is hopeless, because traditional methods require an extremely fine resolution in time and space, which entails unacceptable computational costs. A possible alternative is to replace the original problem by a new system of PDEs which is more suitable for numerical computations but still yields a sufficiently accurate approximation. Such models are often based on the \emph{slowly varying envelope approximation} or generalizations thereof. Results in the literature state that the error of the slowly varying envelope approximation is of $\mathcal{O}(\varepsilon)$. In this work, however, we prove that the error is even proportional to $\varepsilon^2$, which is a substantial improvement, and which explains the error behavior observed in numerical experiments. For a higher-order generalization of the slowly varying envelope approximation we improve the error bound from $\mathcal{O}(\varepsilon^2)$ to $\mathcal{O}(\varepsilon^3)$. Both proofs are based on a careful analysis of the nonlinear interaction between oscillatory and non-oscillatory error terms, and on \textit{a priori} bounds for certain ``parts'' of the approximations which are defined by suitable projections